Large sets with small doubling modulo p are well covered by an arithmetic progression

TL;DR

The paper proves that subsets of integers modulo a prime with small doubling are contained in an arithmetic progression, under minimal size restrictions, extending understanding of additive structure in finite fields.

Contribution

It establishes that small-doubling subsets modulo p are contained in arithmetic progressions without size restrictions, a novel result in additive combinatorics.

Findings

Subsets with small doubling are contained in arithmetic progressions.

The result applies to all sufficiently large primes.

No restrictions on the size of the subset S.

Abstract

We prove that there is a small but fixed positive integer e such that for every prime larger than a fixed integer, every subset S of the integers modulo p which satisfies |2S|<(2+e)|S| and 2(|2S|)-2|S|+2 < p is contained in an arithmetic progression of length |2S|-|S|+1. This is the first result of this nature which places no unnecessary restrictions on the size of S.